Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $n \neq 0$. $p = \dfrac{4n}{2(3n - 1)} \div \dfrac{n}{21n - 7} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{4n}{2(3n - 1)} \times \dfrac{21n - 7}{n} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ 4n \times (21n - 7) } { 2(3n - 1) \times n } $ $ p = \dfrac {4n \times 7(3n - 1)} {n \times 2(3n - 1)} $ $ p = \dfrac{28n(3n - 1)}{2n(3n - 1)} $ We can cancel the $3n - 1$ so long as $3n - 1 \neq 0$ Therefore $n \neq \dfrac{1}{3}$ $p = \dfrac{28n \cancel{(3n - 1})}{2n \cancel{(3n - 1)}} = \dfrac{28n}{2n} = 14 $